A Bloch transform based numerical method for the rough surface scattering problems

TL;DR

This paper introduces a novel numerical method based on the Bloch transform for solving rough surface scattering problems, demonstrating convergence and potential for high-order extensions.

Contribution

It develops a new approach using Bloch transform techniques to approximate and solve rough surface scattering problems with proven convergence.

Findings

Finite Fourier series approximation of the Bloch transformed field is exact for truncated surfaces.

The proposed numerical method converges and is validated by numerical experiments.

The approach offers a new perspective and potential for high-order methods in rough surface scattering.

Abstract

In this paper, we will study the Bloch transformed rough surface scattering problems, and propose a numerical method based on the Bloch transformed problems. Based on the mathematical theory of the scattering problems from locally perturbed periodic surfaces, the same techniques will be applied to the rough surface scattering problems, and an equivalent coupled family of quasi-periodic scattering problems in one periodic cell will be established. The most important result obtained in this paper is on the finite Fourier series approximation of the Bloch transformed field with respect to the quasi-periodicity parameter. It will be proved that the finite series is exactly the Bloch transformed solution corresponds to truncated rough surfaces. Thus the truncation provides a reasonable approximation, and could be applied to the numerical solutions. Based on the approximation, a numerical method is proposed for the rough surface scattering problems. The convergence of the numerical method is proved and illustrated by the numerical experiments. The method provides a completely new perspective for the rough surface scattering problems. There is possibility that some high order method will be developed based on this new method.